On Generalized Rickart $*$-rings

Abstract: A ring $R$ with an involution $$ is a generalized Rickart $$-ring if for all $x\in R$ the right annihilator of $x^n$ is generated by a projection for some positive integer $n$ depending on $x$. In this work, we introduce generalized right projection of an element in a $$-ring and prove that every element in a generalized Rickart $$-ring has generalized right projection. Various characterizations of generalized Rickart $$-rings are obtained. We introduce the concept of generalized weakly Rickart $$-ring and provide a characterization of generalized Rickart $$-rings in terms of weakly generalized Rickart $$-rings. It is shown that generalized Rickart $$-rings satisfy the parallelogram law. A sufficient condition is established for partial comparability in generalized Rickart $$-rings. Furthermore, it is proved that pair of projections in a generalized Rickart $*$-ring possess orthogonal decomposition.